1 Definicja
R2 = {(x, y) : x, y " R} .
2 Definicja r P = (a, b)
O(P ) = {(x, y) : (x - a)2 + (y - b)2 < r}.
r P = (a, b)
S(P ) = O(P ) \ {P }
3 Definicja f A Ä…" R2 R
A
f : A R z = f(x, y) (x, y) " A f (x, y)
f(x, y) A D(f)
4 PrzykÅ‚ad f : R × R R
Å„Å‚
òÅ‚0
y d" x
f(x, y) =
1
ół
(y - x) y > x
2
g : R × R {-1, 0, 1}
Å„Å‚
ôÅ‚
òÅ‚1 x y
g(x, y) = 0 x y
ôÅ‚
ół-1
h : R × R R h(x, y) = y(x2 - 1)
f(x, y) =
"
xy
D(f) = {(x, y) " R2 : (x e" 0 y e" 0) (x d"
0 y d" 0)}
5 Definicja f R3
z = f(x, y)
{(x, y, z) : (x, y) " D(f), z = f(x, y)}.



6 Definicja (x0, y0)
(x0, y0) z0 %EÅ‚ > 0 ´ > 0

(x, y) (x0, y0) (x - x0)2 + (y - y0)2 <
´ |f(x, y) - z0| < %EÅ‚
(x0, y0)
7 Przykład
Å„Å‚
3
òÅ‚x + y3
, (x, y) = (0, 0),

2
f(x, y) =
ółx + y2
0, (x, y) = (0, 0).
(0, 0) 0 %EÅ‚ > 0 (x, y) = (0, 0)


3

x + y3 x3
|f(x, y) - 0| < %EÅ‚ Ð!Ò!
x2 + y2 - 0 < %EÅ‚ Ð!Ò! + y3 < %EÅ‚ (x2 + y2)


Ð!= |x3| < %EÅ‚ · x2 '" |y3| < %EÅ‚ · y2 Ð!Ò! |x| < %EÅ‚ '" |y| < %EÅ‚ Ð!= x2 + y2 < %EÅ‚.
´ = %EÅ‚

Å„Å‚
2
òÅ‚x - y2
, (x, y) = (0, 0),

2
f(x, y) =
ółx + y2
0, (x, y) = (0, 0).
(0, 0) (x, y) = (a, a) a = 0

f(x, y) = 0 (x, y) = (a, 0) a = 0 f(x, y) = 1

´ > 0 ´ (0, 0) 1
1
0 z0 |f(x, y) - z0| <
3
8 Definicja P : N R2
n n
Pn = (xn, yn) (Pn) ((xn, yn))
9 Definicja (Pn) = ((xn, yn)) P0 = (x0, y0)
lim xn = x0 lim yn = y0.
n" n"

lim Pn = P0 lim (xn, yn) = (x0, y0).
n" n"

(-1)n
10 Przykład 1, 1 + (1, 1)
n
(1, 1 + (-1)n)

11 Definicja (x0, y0) " R2 f
S(x0, y0) z0 f (x0, y0) ((xn, yn)) Ä…"
S(x0, y0)
lim (xn, yn) = (x0, y0) =Ò! lim f(xn, yn) = z0.
n" n"

lim = z0.
(x,y)(x0,y0)

12 Przykład
Å„Å‚
3
òÅ‚x + y3
, (x, y) = (0, 0),

2
f(x, y) =
ółx + y2
0, (x, y) = (0, 0).
(0, 0) 0 ((xn, yn))
(0, 0)
3 3
x3 yn x3 yn
n n
- 0 '" - 0 =Ò! - 0 '" - 0
2 2 2
x2 yn x2 + yn x2 + yn
n n n
3
x3 + yn
n
=Ò! - 0 Ð!Ò! f(xn, yn) - 0.
2
x2 + yn
n

Å„Å‚
2
òÅ‚x - y2
, (x, y) = (0, 0),

2
f(x, y) =
ółx + y2
0, (x, y) = (0, 0).

1 1
(0, 0) (xn, yn) = , (0, 0)
n n


1 1
lim f , = lim 0 = 0.
n" n"
n n

1
(xn, yn) = , 0
n

1
lim f , 0 = lim 1 = 1 = 0.

n" n"
n
13 Definicja (x0, y0)

14 Definicja M


15 TWIERDZENIE




16 Definicja
(x0, y0)
f(x0 + "x, y0) - f(x0, y0)
lim .
"x0
"x
"f
2
(x0, y0) fx(x0, y0) fx(x0, y0)
"x
(x0, y0)

f(x0, y0 + "y) - f(x0, y0)
lim .
"y0
"y
"f
2
(x0, y0) fy(x0, y0) fy(x0, y0)
"y
x
y
y x


17 Przykład x2 + y2 = r2 cos Ć = x/r
tg Ć = y/x
Å„Å‚
òÅ‚1
sin 2Ć, r = 0

f(r, Ć) =
r
ół0,
r = 0
Ä„
r = 0 Ć =
4

Ä„ 1 Ä„ 1
lim f r, = lim sin = lim = ".
r0 r0 r0
4 r 2 r
Ä„ 3Ä„
x y = 0 Ć = Ć =
2 2
sin 2Ć = 0 f(x, 0) = 0 fx(0, 0) = 0 y
f(x, y) = |y| R2 fx(1, 0) = 0
y




18 TWIERDZENIE

"(f + g) "f "g
(x0, y0) = (x0, y0) + (x0, y0),
"x "x "x
"(f - g) "f "g
(x0, y0) = (x0, y0) - (x0, y0),
"x "x "x
"(f · g) "f "g
(x0, y0) = (x0, y0) · g(x0, y0) + f(x0, y0) · (x0, y0),
"x "x "x
"f "g
"(f/g) (x0, y0) · g(x0, y0) - f(x0, y0) · (x0, y0)
"x "x
(x0, y0) = .
"x g2(x0, y0)

19 Definicja f
D Ä…" R2
"f "f
(x, y) (x, y) (x, y) ,
"x "y
(x, y) " D f D
"f "f
2 2
fx fy fx fy
"x "y
20 Przykład f(x, y) = x2y3 - x sin y
"f "f
(x, y) = 2xy3 - sin y, (x, y) = 3x2y2 - x cos y.
"x "y
g(x, y) = x5y10 - x3 sin y + y2ex
"g "g
(x, y) = 5x4y10 - 3x2 sin y + y2ex, (x, y) = 10x5y9 - x3 cos y + 2yex.
"x "y
n
Rn


21 Przykład f(x, y, z) = x2y3z4 - y sin z
g(x, y, z) = x5y10z - z sin y + y2ez

"f "f
22 Definicja
"x "y
f

"2f " "f
fxx = = ,
"x2 "x "x

"2f " "f
fxy = = ,
"x"y "x "y

"2f " "f
fyx = = ,
"y"x "y "x

"2f " "f
fyy = = .
"y2 "y "y


23 Przykład f(x, y) = x2y3 - x sin y

"2f " "f "
(x, y) = (x, y) = (2xy3 - sin y) = 2y3
"x2 "x "x "x

"2f " "f "
(x, y) = (x, y) = (3x2y2 - x cos y) = 6xy2 - cos y
"x"y "x "y "x

"2f " "f "
(x, y) = (x, y) = (2xy3 - sin y) = 6xy2 - cos y
"y"x "y "x "y

"2f " "f "
(x, y) = (x, y) = (3x2y2 - x cos y) = 6x2y + x sin y
"y2 "y "y "y
g(x, y) = x5y10 - x3 sin y + y2ex

"2g " "g "
(x, y) = (x, y) = (5x4y10 - 3x2 sin y + y2ex)
"x2 "x "x "x
= 20x3y10 - 6x sin y + y2ex

"2g " "g "
(x, y) = (x, y) = (10x5y9 - x3 cos y + 2yex)
"x"y "x "y "x
= 50x4y9 - 3x2 cos y + 2yex

"2g " "g "
(x, y) = (x, y) = (5x4y10 - 3x2 sin y + y2ex)
"y"x "y "x "y
= 50x4y9 - 3x2 cos y + 2yex

"2g " "g "
(x, y) = (x, y) = (10x5y9 - x3 cos y + 2yex)
"y2 "y "y "y
= 90x5y8 + x3 sin y + 2ex
24 Definicja
x y



25 TWIERDZENIE f M
(x0, y0) fxy fyx (x0, y0)

fxy(x0, y0) = fyx(x0, y0).
26 Przykład


f (x0, y0)
f y = y0
(x0, y0, f(x0, y0)) Ä… OXY

"f
(x0, y0) = tg Ä….
"x
f x = x0
(x0, y0, f(x0, y0)) ²
OXY
"f
(x0, y0) = tg ².
"y